• Corpus ID: 250311928

# Critical Window of The Symmetric Perceptron

@inproceedings{Altschuler2022CriticalWO,
title={Critical Window of The Symmetric Perceptron},
author={Dylan J. Altschuler},
year={2022}
}
We study the critical window of the symmetric binary perceptron, or equivalently, combinatorial discrepancy. Consider the problem of finding a binary vector σ satisfying ‖Aσ‖∞ ≤ K, where A is an αn×n matrix with iid Gaussian entries. For fixed K, at which densities α is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu [28], and Abbe, Li, and Sly [2], answering this to first order. Namely, for each K there exists an explicit…

## References

SHOWING 1-10 OF 38 REFERENCES

• Mathematics, Computer Science
STOC
• 2019
This paper shows that the Krauth–Mézard conjecture α⋆ is a lower bound with positive probability, under the condition that an explicit univariate function S(λ) is maximized at λ=0.83.
• Computer Science
2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
• 2022
It is shown that at high enough densities the SBP exhibits the multi Overlap Gap Property (m-OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms.
• Mathematics
2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
• 2022
We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory
• Computer Science
STOC
• 2021
It is proved, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure; that is, typical solutions of the model lie in clusters of vanishing entropy density.
• Computer Science, Mathematics
COLT
• 2020
A randomized polynomial-time algorithm is presented that achieves discrepancy $e^{-\Omega(\log^2(n)/m)}$ with high probability, provided that $m \leq O(\sqrt{\log{n}})$.
• Computer Science
Journal of Physics A: Mathematical and Theoretical
• 2019
The replica method is used to estimate the capacity threshold for the rectangle-binary-perceptron case when the u-function is wide and it is concluded that full-step-replica-symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case.
• Computer Science
STOC
• 2022
It is shown that at low constraint density, there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu in STOC'21.
• Mathematics
• 2011
We study integer programming instances over polytopes P(A,b)={x:Ax<=b} where the constraint matrix A is random, i.e., its entries are i.i.d. Gaussian or, more generally, its rows are i.i.d. from a
• Mathematics
ArXiv
• 2021
This work applies the partial colouring lemma of Lovett and Meka to show that w.h.p. has discrepancy O( √ dn/m log(m/n), and characterizes how the discrepancy of each random hypergraph model transitions from â‚ d to o(√ d) as m varies from m = Θ(n) to m n.
Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Knl/2, K an absolute constant. This improves the basic probabilistic method with